When using the MaxSPRT and the CV.Poisson function to conduct continuous sequential analysis for Poisson data, the null
hypothesis is rejected when the log likelihood ratio exceeds the pre-determined critical value calculated by CV.Poisson.
The sequential analysis ends without rejecting the null hypothesis when a predetermined upper limit on the sample size is
reached, expressed in terms of the expected number of events under the null hypothesis. For example, the sequential analysis
may end as soon as the sample size is such that there are 50 expected events under the null.
The default in the function SampleSize.Poisson is for calculating the upper limit on the sample size (length of surveillance) required
for the continuous Poisson based MaxSPRT (alphaSpend="n") to achieve the desired statistical power for a pre-specified relative risk RR.
The solution is exact using iterative numerical calculations (Kulldorff et al., (2011).
While designed for continuous sequential analysis with flat threshold in the scale of the MaxSPRT statistic, the SampleSize.Poisson function can also be used to approximate the
required upper limit on the sample size that is needed when doing group sequential analysis for Poisson data, using the CV.G.Poisson function.
There is also the possibility to calculate the sample size for an user-defined alpha spending plan. This is possible with the input parameter alphaSpend.
The user can select among one of the four classical alpha spending shapes bellow:
\(F_{1}(t)=\alpha t^{\rho}\), where \(\rho>0\),
\(F_{2}(t)=2-2\Phi(x_{\alpha}\sqrt{t^{-1}})\), where \(x_{\alpha}=\Phi^{-1}(1-\alpha/2)\),
\(F_{3}(t)= \alpha \times log(1+[exp{1}-1]\times t) \),
\(F_{4}(t)=\alpha[1-exp(-t\gamma)]/[1-exp(-\gamma)]\) with \(\gamma \in \Re\),
and \(t\) represents a fraction of the maximum length of surveillance.
To select one of the four alpha spending types above, and using an integer \(i\) to indicate the type among
\(i=\) 1, 2, 3, and 4, for \(F_{1}(t)\), \(F_{2}(t)\), \(F_{3}(t)\) and \(F_{4}(t)\), respectively,
one needs to set alphaSpend=i. Specifically for alphaSpend=1, it is necessary to choose a rho value,
or a gamma value if alphaSpend=4 is used.
For more details on these alpha spending choices,
see the paper by Silva et al. (2021), Section 2.7.
When one sets alphaSpend=i, the threshold impplied by the correspondent alpha spending is calculated.
The function delivers the threshold in the scale of a test statistic selected by the user with the input
Statistic among the classic methods:
MaxSPRT (Kulldorf et al., 2011), Pocock (Pocock, 1977), OBrien-Fleming (O'Brien and Fleming, 1979), or Wang-Tsiatis (Jennison and Turnbull, 2000).
For Statistic="Wang-Tsiatis", the user has to choose a number in the (0, 0.5] interval for Delta.